Let XH denote the number of copies of a fixed graph H in Gn,p. Gilmer and Kopparty conjectured that XH
Nearby in the stack
satisfies a local central limit theorem (LCLT) provided that
H
is connected,
p≫n−1/m(H)
, and
n2(1−p)≫1
, where
m(H)
is the maximum density. Following the work of Berkowitz, Sah and Sawhney confirmed this conjecture for every constant
p
, leaving the regime where
p=o(1)
open. In this regime, the only case addressed in the literature is when
H=K3
, where, in a recent paper, Araújo and Mattos confirmed the conjecture for
p∈(4n−1/2,1/2)
. This, together with a general result of Röllin and Ross, essentially settles the conjecture for the triangle. We generalise these results by showing that an LCLT holds for
H=Kr
(for any fixed
r≥3
) in the regime
n−1/m(H)≪p≤1/2
, essentially settling the conjecture for cliques.